We show that pseudospin symmetry in nuclei could arise from nucleons moving in a relativistic mean field which has an attractive scalar and repulsive vector potential nearly equal in magnitude.[S0031-9007(96)02176-X] PACS numbers: 21.60. Cs, 21.30.Fe, 24.10.Jv Almost 30 years ago a quasidegeneracy was observed in heavy nuclei between single-nucleon doublets with quantum numbers ͑n r , ᐉ, j ᐉ 1 1 2 ͒ and ͑n r 2 1, ᐉ 1 2, j ᐉ 1 3 2 ͒ where n r , ᐉ, and j are the single nucleon radial, orbital, and total angular momentum quantum numbers, respectively [1,2]. These authors defined a "pseudo" orbital angular momentumᐉ ᐉ 1 1; for example, ͑ ͑ ͑n r s 1͞2 ͑n r 2 1͒d 3͞2 ͒ ͒ ͒ will haveᐉ 1, ͑ ͑ ͑n r p 3͞2 , ͑n r 2 1͒f 5͞2 ͒ ͒ ͒ will haveᐉ 2, etc. Then these doublets are almost degenerate with respect to "pseudo" spin,s 1 2 , since j ᐉ 6s for the two states in the doublet. This symmetry has been used to explain a number of phenomena in nuclear structure [3] including most recently the identical rotational bands observed in nuclei [4]. Despite this long history of pseudospin symmetry [5,6], the origin of this symmetry has eluded explanation. Recently it was shown [7] that relativistic mean field theories predict the correct spin-orbit splitting [8]. In this paper we identify a possible reason for this; namely that the symmetry arises from the near equality in magnitude of an attractive scalar, 2V s , and repulsive vector, V y , relativistic mean fields, V s ϳ V y , in which the nucleons move. Such a near equality of mean fields follows from relativistic field theories with interacting nucleons and mesons [8], with nucleons interacting via Skyrme-type interactions [9], and from QCD sum rules [10].A nucleon moving in a spherical field has the total angular momentum j, its projection on the z axis, m, and k 2b͑ŝ ?L 1 1͒ conserved, whereb is the Dirac matrix [11]. The eigenvalues ofk are k 6͑j 1 1 2 ͒; 2 for aligned spin ͑s 1͞2 , p 3͞2 , etc.͒ and 1 for unaligned spin ͑p 1͞2 , d 3͞2 , etc.͒. Hence we use the quantum number k since it is sufficient to label the orbitals. The Dirac equation for the single-nucleon radial wave function ͑g k , f k ͒ in dimensionless units is given by [11] ∑ d drwhere r is the radial coordinate in units of length hc͞mc 2 , V ͑r͒ ͓V y ͑r͒ 1 V s ͑r͔͒͞mc 2 , D͑r͒ ͓V s ͑r͒ 2 V y ͑r͔͒͞mc 2 , and E is the binding energy ͑E . 0͒ of the nucleon in units of the nucleon mass mc 2 . First we show that, in the limit of equality of the magnitude of the vector and scalar potential D͑r͒ 0, pseudospin is exactly conserved. To do this we solve for g k in (2) and substitute into (1), obtaining the second order differentialwhich agrees with the original definition of the pseudoorbital angular momentum [1,2]. For example, for ͓n r s 1͞2 , ͑n r 2 1͒d 3͞2 ͔, k 21 and 2, respectively, givingᐉ 1 in both cases. Futhermore, the physical significance ofᐉ is revealed; it is the "orbital angular momentum" of the lower component of the Dirac wave function. Eq.(3) is a Schrödinger equation with an attractive potential V and bind...
